Travelling wave solutions of the reaction-diffusion mathematical model of glioblastoma growth: An Abel equation based approach

We consider quasi-stationary (travelling wave type) solutions to a nonlinear
reaction-diffusion equation with arbitrary, autonomous coefficients, describing
the evolution of glioblastomas, aggressive primary brain tumors that are
characterized by extensive infiltration into the brain and are highly resistant
to treatment. The second order nonlinear equation describing the glioblastoma
growth through travelling waves can be reduced to a first order Abel type
equation. By using the integrability conditions for the Abel equation several
classes of exact travelling wave solutions of the general reaction-diffusion
equation that describes glioblastoma growth are obtained, corresponding to
different forms of the product of the diffusion and reaction functions. The
solutions are obtained by using the Chiellini lemma and the Lemke
transformation, respectively, and the corresponding equations represent
generalizations of the classical Fisher--Kolmogorov equation. The biological
implications of two classes of solutions are also investigated by using both
numerical and semi-analytical methods for realistic values of the biological
parameters.

Angiogenesis involves the formation of new blood vessels by sprouting or
splitting of existing blood vessels. During sprouting, a highly motile type of
endothelial cell, called the tip cell, migrates from the blood vessels followed
by stalk cells, an endothelial cell type that forms the body of the sprout. In
vitro models and computational models can recapitulate much of the
phenomenology of angiogenesis in absence of tip and stalk cell differentiation.
Therefore it is unclear how the presence of tip cells contributes to
angiogenesis. To get more insight into how tip cells contribute to
angiogenesis, we extended an existing computational model of vascular network
formation based on the cellular Potts model with tip and stalk differentiation,
without making a priori assumptions about the specific rules that tip cells
follow. We then screened a range of model variants, looking for rules that make
tip cells (a) move to the sprout tip, and (b) change the morphology of the
angiogenic networks. The screening predicted that if tip cells respond less
effectively to an endothelial chemoattractant than stalk cells, they move to
the tips of the sprouts, which impacts the morphology of the networks. A
comparison of this model prediction with genes expressed differentially in tip
and stalk cells revealed that the endothelial chemoattractant Apelin and its
receptor APJ may match the model prediction. To test the model prediction we
inhibited Apelin signaling in our model and in an in vitro model of angiogenic
sprouting, and found that in both cases inhibition of Apelin or of its receptor
APJ reduces sprouting. Based on the prediction of the computational model, we
propose that the differential expression of Apelin and APJ yields a
"self-generated" gradient mechanisms that accelerates the extension of the
sprout.

Glucose-lactate metabolic cooperation in cancer: insights from a spatial mathematical model and implications for targeted therapy

ABSTRACT

A recent hypothesis has proposed a glucose-lactate metabolic symbiosis between adjacent hypoxic and oxygenated regions of a developing tumour, and proposed a treatment strategy to target this symbiosis. However, in vivo experimental support remains inconclusive. Here we develop a minimal spatial mathematical model of glucose-lactate metabolism to examine, in principle, whether metabolic symbiosis is plausible in human tumours, and to assess the potential impact of inhibiting it. We find that symbiosis is a robust feature of our model system---although on the length scale at which oxygen supply is diffusion-limited, its occurrence requires very high cellular metabolic activity---and that necrosis in the tumour core is reduced in the presence of symbiosis. Upon simulating therapeutic inhibition of lactate uptake, we predict that targeted treatment increases the extent of tissue oxygenation without increasing core necrosis. The oxygenation effect is correlated strongly with the extent of wildtype hypoxia and only weakly with wildtype symbiotic behaviour, and therefore may be promising for radiosensitisation of hypoxic, lactate-consuming tumours even if they do not exhibit a spatially well-defined symbiosis. Finally, we conduct a set of in vitro experiments on the U87 glioblastoma cell line to facilitate preliminary speculation as to where highly malignant tumours might fall in our parameter space, and find that these experiments suggest a weakly symbiotic regime for U87 cells, which raises the new question of what relationship exists between symbiosis---if indeed it occurs in vivo---and tumour malignancy.